Paper · Mar 30, 2026
A note on Ricci flow from small curvature concentration and a Morrey-type condition
As on arXiv
Abstract
In the authors proved the long-time existence of Ricci flow starting from complete bounded curvature Riemannian manifolds with scale-invariant integral curvature bounded by a dimensional constant times the inverse of the Sobolev constant. We generalize this result by replacing the bounded curvature assumption with the assumption that is only equivalent to a complete bounded curvature metric while satisfying a Morrey-type condition on the gradient of relative to : a local integral condition on the covariant derivative . The Morrey-type condition was first considered in in the context of Ricci flow on non-compact manifolds, and in particular allows the possibility for to have unbounded curvature on . As in, our long-time solution enjoys curvature decay estimates implying in particular that is diffeomorphic to .
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
- 11A.1The Equation and Its First Solutions
A heat equation for the metric: spheres, cylinders and scaling.